CHAPTER01ARTICLE04

FLUID MECHANICS · 01–04 / ADVANCED

Eulerian & Lagrangian descriptions

Connect the fixed-point Eulerian and particle-following Lagrangian descriptions of a flow through the flow map and material derivative.

10 min read2026-08-13Definitions, equations & units checkedJA version

Abstract

The Eulerian description represents velocity and pressure as fields in space and time; the Lagrangian description follows individual particles. They describe the same motion with different independent variables, and the material derivative connects field changes to changes experienced by a moving particle.

Eulerian description: fix a point in space

The Eulerian description represents quantities as fields, such as density ρ(x, t), pressure p(x, t) and velocity u(x, t). A measurement point, control volume or computational grid remains fixed in space while different fluid particles pass through it.

A pressure transducer mounted at a fixed pipe location and a conventional CFD solution of the Navier–Stokes equations on a fixed grid are Eulerian examples.

Equation (1)Eulerian field

\phi=\phi(\boldsymbol{x},t)

$\phi$: A field such as density, temperature or a velocity component[Unit of φ]
$\boldsymbol{x}$: Fixed spatial position[ $\mathrm{m}$ ]
$t$: Time[ $\mathrm{s}$ ]

Held fixed: Spatial position or control volume
Well suited to: Continuum fields with fixed boundaries, pipe and external flows, fixed-grid CFD

Lagrangian description: fix particle identity

Label each particle by a at an initial time and describe its current position by the flow map X(a, t). Holding a fixed while advancing time gives the history of one particle's position, velocity, temperature and other properties.

Particle tracking, droplets, bubbles, free surfaces and large-deformation problems often have a natural Lagrangian representation. Strong deformation, however, can make neighborhood relations and numerical resolution difficult to maintain.

Equation (2)Flow map and particle velocity

\frac{\mathrm d\boldsymbol{X}(\boldsymbol{a},t)}{\mathrm d t}=\boldsymbol{u}\!\left(\boldsymbol{X}(\boldsymbol{a},t),t\right),\qquad \boldsymbol{X}(\boldsymbol{a},t_0)=\boldsymbol{a}

$\boldsymbol{a}$: Particle label or initial position[ $\mathrm{m}$ ]
$\boldsymbol{X}(\boldsymbol{a},t)$: Current position of particle a[ $\mathrm{m}$ ]
$\boldsymbol{u}$: Velocity field[ $\mathrm{m/s}$ ]

Held fixed: Fluid-particle identity or label
Well suited to: Particle histories, droplets and bubbles, interface tracking, particle methods

Material derivative: connecting the two descriptions

Observe an Eulerian field φ(x, t) along a moving particle x = X(a, t). Differentiating the composite function φ(X(a, t), t) and using dX/dt = u gives the material derivative [1, 2].

The first term on the right is the local time change at a fixed point. The second is the advective change caused by motion through a spatial gradient. Even when ∂φ/∂t = 0 in steady flow, a particle can experience change if the field is spatially nonuniform.

Equation (3)Material derivative of a scalar field

\frac{\mathrm D\phi}{\mathrm D t}=\frac{\partial\phi}{\partial t}+\boldsymbol{u}\cdot\nabla\phi

$\mathrm D\phi/\mathrm D t$: Rate experienced by a moving particle[ $\phi/\mathrm{s}$ ]
$\partial\phi/\partial t$: Local rate at a fixed point[ $\phi/\mathrm{s}$ ]
$\boldsymbol{u}\cdot\nabla\phi$: Advective rate of change[ $\phi/\mathrm{s}$ ]

Equation (4)Fluid-particle acceleration

\boldsymbol{a}=\frac{\mathrm D\boldsymbol{u}}{\mathrm D t}=\frac{\partial\boldsymbol{u}}{\partial t}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}

$\boldsymbol{a}$: Particle acceleration[ $\mathrm{m/s^2}$ ]
$\partial\boldsymbol{u}/\partial t$: Local acceleration[ $\mathrm{m/s^2}$ ]
$(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$: Advective acceleration[ $\mathrm{m/s^2}$ ]

Choosing and combining the descriptions

Eulerian and Lagrangian descriptions do not represent different physics; they use different coordinates for the same motion. Eulerian variables are direct for conservation laws over fixed control volumes, while Lagrangian variables are direct for the histories of individual material elements.

Engineering models often combine them. In an Euler–Lagrange calculation, for example, air is solved as a continuum on an Eulerian grid while dispersed droplets are tracked as Lagrangian particles. The choice follows the quantity to be conserved or tracked, boundary motion and computational constraints.

Comparison of the two descriptions
Aspect	Eulerian	Lagrangian
Independent variables	Position x and time t	Particle label a and time t
Observation	Particles passing a fixed point	History of the same particle
Representative methods	Finite volume and fixed-grid methods	Particle and moving-mesh methods
Typical challenge	Advection and numerical diffusion of interfaces	Large deformation of particles or meshes

References

Sources accessed 13 August 2026. Independent expert review has not yet been completed.